Is there a hotkey for automatic key-frame insertion? If not how do I set it?
There is no such a key as far as I know.
2.79
To add it in 2.7 you should go to User Preferences(ctrl+alt+u), Input tab:

And add a new hotkey for the area of Blender you would want it to work in. You can use wm.context_toggle with a context attribute tool_settings.use_keyframe_insert_auto:

2.8
In 2.8 you can just right click the icon and choose Assign Shortcut:

It seems the button does not refresh instantly on the interface when you use such a shortcut. It refreshes when you move the mouse to another editor or perform some other action that refreshes the UI. I believe that should not affect the functionality.